Write Expressions With Numbers and Variables - 6th Grade Math

Does the expression $x-7$ represent the phrase "7 less than a number $x$ "?
Answer: Yes, because it subtracts 7 from $x$ . This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). Example: "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct answer is yes, the expression x - 7 represents "7 less than a number x" because it subtracts 7 from x. A common error is reversing the order, thinking it should be 7 - x, or confusing with addition like x + 7 or multiplication. Writing: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $m$ represent a number of minutes. Which expression means half of $m$ ?
Answer: $\dfrac{m}{2}$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: 'more than/sum' → addition (+), 'less than/difference' → subtraction (-), 'times/product' → multiplication (coefficient: $3x$ means 3 times x), 'divided by/quotient' → division ( $x/4$ ). Order matters for non-commutative: '5 less than n' means subtract 5 FROM n ( $n-5$ , not $5-n$ reversed), 'n divided by 4' means $n/4$ (not $4/n$ ). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, '7 more than a number' uses variable n for number, 'more than' means add, expression: $n+7$ ; 'product of 3 and x' means 3 times x: $3x$ ; 'twice a number plus 5' means 2 times n plus 5: $2n+5$ (not $2(n+5)$ which would be twice the sum); 'x divided by 8' means $x/8$ . The correct translation for 'half of m' is $m/2$ , which matches choice C. A common error is reversing the division to $2/m$ , or confusing with multiplication to get $2m$. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for 'a number'), (3) determine order (for subtraction/division: 'less than' and 'divided by' require careful order—'5 less than n' is $n-5$ , subtract from first quantity), (4) write expression ( $n+5$ , $3x$ , $x-7$ , $y/4$ ), (5) verify (if n=10: '5 more than n' gives 10+5=15✓, makes sense). Key phrases: 'more than' adds to variable ( $n+5$ ), 'less than' subtracts from variable ( $n-7$ ), 'times' multiplies ( $3n$ coefficient notation), 'of' often means multiply (half of n: $(1/2)n=n/2$ ). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of $3x$ , acceptable but coefficient notation preferred).
A rectangular garden has a length that is $4$ meters longer than twice its width. If the width is $w$ meters, which expression represents the perimeter of the garden?
Answer: $6w + 8$ The width is $w$ and the length is $2w + 4$ . The perimeter formula is $2(\text{length} + \text{width}) = 2(2w + 4 + w) = 2(3w + 4) = 6w + 8$ . Choice B uses the incorrect perimeter formula $2\text{length} + 2\text{width}$ but makes an error. Choice C represents length plus width, not perimeter. Choice D represents just the length.
A taxi company charges a base fee of $2.50 plus $0.75 for each quarter-mile traveled. If a passenger travels $m$ quarter-miles, which expression represents the total cost of the trip?
Answer: $2.50 + 0.75m$ When you see a word problem about costs with both a fixed fee and a variable rate, you're dealing with linear expressions. The key is identifying what stays the same (fixed cost) versus what changes based on usage (variable cost). Let's break down this taxi problem step by step. The company charges $2.50 as a base fee - this is what you pay no matter how far you travel. Then they add $0.75 for each quarter-mile you go. If you travel $m$ quarter-miles, you'll pay $0.75 \times m$ dollars for the distance portion. Your total cost is the base fee plus the distance charge: $2.50 + 0.75m$ . Looking at the wrong answers: Choice A ( $3.25m$ ) incorrectly combines the base fee and per-quarter-mile rate into one coefficient, eliminating the fixed cost structure entirely. Choice C ( $2.50m + 0.75$ ) swaps the fixed and variable parts - it treats the base fee as if it multiplies by distance and the per-quarter-mile rate as fixed. Choice D ( $0.75(2.50 + m)$ ) uses the distributive property incorrectly, suggesting you pay $0.75 \times 2.50$ plus $0.75 \times m$ , which doesn't match the problem's structure. The correct answer is B: $2.50 + 0.75m$ . Study tip: In cost problems, always identify the fixed cost first (what you pay regardless), then add the variable cost (rate × quantity). The pattern is: Fixed Cost + (Rate × Variable).
Let $n$ be a number. Which expression represents 2 more than twice $n$ ?
Answer: $2n+2$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for '2 more than twice n' is 2n + 2, which matches choice A. A common error is applying the 'more than' to the whole, like writing 2(n + 2) instead of 2n + 2, or confusing with subtraction to get something like n - 2. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A video game gives you $x$ points, then you lose 7 points. Which expression represents 7 less than $x$ ?
Answer: $x-7$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for '7 less than x' is x - 7, which matches choice A. A common error is reversing the order, like writing 7 - x instead of x - 7, or confusing it with addition to get x + 7. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A notebook costs $x$ dollars. A pen costs $2$ dollars more than the notebook. Which expression represents the cost of the pen?
Answer: $x+2$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols ( $+$ , $-$ , $\times$ , $\div$ ). Translating words to algebra: "more than/sum" → addition ( $+$ ), "less than/difference" → subtraction ( $-$ ), "times/product" → multiplication (coefficient: $3x$ means 3 times x), "divided by/quotient" → division ( $x/4$ ). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n ( $n-5$ , not $5-n$ reversed), "n divided by 4" means $n/4$ (not $4/n$ ). Variable represents number: x, n, y are placeholders (unknown or any number in context). Example: "7 more than a number" uses variable n for number, "more than" means add, expression: $n+7$ ; "product of 3 and x" means 3 times x: $3x$ ; "twice a number plus 5" means 2 times n plus 5: $2n+5$ (not $2(n+5)$ which would be twice the sum); "x divided by 8" means $x/8$ . The correct translation for the cost of the pen which is "2 dollars more than the notebook" where the notebook costs x dollars is $x + 2$ . A common error is confusing "more than" with multiplication like $2x$ , subtraction like $x - 2$ , or division like $x/2$ . Writing: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is $n-5$ , subtract from first quantity), (4) write expression ( $n+5$ , $3x$ , $x-7$ , $y/4$ ), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable ( $n+5$ ), "less than" subtracts from variable ( $n-7$ ), "times" multiplies ( $3n$ coefficient notation), "of" often means multiply (half of n: $(1/2)n=n/2$ ). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing $3\times x$ instead of $3x$ , acceptable but coefficient notation preferred).
A ribbon is $y$ centimeters long. It is cut into 4 equal pieces. Which expression gives the length of one piece?
Answer: $\dfrac{y}{4}$ This problem is all about sharing equally. When you split something into equal pieces, you divide. Here, the ribbon is $y$ centimeters long, and you cut it into $4$ equal pieces. To find the length of one piece, divide the total length by the number of pieces: $\frac{y}{4}$ That matches choice D. Think about it like a pizza. If a pizza weighs $y$ ounces and you slice it into $4$ equal slices, each slice weighs $\frac{y}{4}$ ounces. The total goes on top, and the number of pieces goes on the bottom. So whenever you see "cut into equal pieces" or "shared equally," reach for division! Try this at home: Grab a string or piece of paper. Pretend it's $y$ long, fold it into $4$ equal parts, and say out loud: "Each part is $\frac{y}{4}$ !" Try it with $3$ parts and $5$ parts too.
A classroom has $x$ students. 7 fewer than that number join an after-school club. Which expression represents the number who join?
Answer: $x-7$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for '7 fewer than x' is x - 7, which matches choice C. A common error is reversing the subtraction, like writing 7 - x, or using addition to get x + 7. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A ribbon is $x$ centimeters long. It is cut into 4 equal pieces. Which expression represents $x$ divided by 4?
Answer: $\dfrac{x}{4}$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for 'x divided by 4' is x/4, which matches choice C. A common error is reversing the division, like writing 4/x, or confusing it with multiplication to get 4x. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $x$ represent a number. Which expression means the sum of twice $x$ and 7?
Answer: $2x+7$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: 'more than/sum' → addition (+), 'less than/difference' → subtraction (-), 'times/product' → multiplication (coefficient: 3x means 3 times x), 'divided by/quotient' → division (x/4). Order matters for non-commutative: '5 less than n' means subtract 5 FROM n (n-5, not 5-n reversed), 'n divided by 4' means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, '7 more than a number' uses variable n for number, 'more than' means add, expression: n+7; 'product of 3 and x' means 3 times x: 3x; 'twice a number plus 5' means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); 'x divided by 8' means x/8. The correct translation for 'the sum of twice x and 7' is 2x + 7, which matches choice A. A common error is misinterpreting the grouping, like writing 2(x + 7) which means twice the sum instead of sum of twice x and 7. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for 'a number'), (3) determine order (for subtraction/division: 'less than' and 'divided by' require careful order—'5 less than n' is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: '5 more than n' gives 10+5=15✓, makes sense). Key phrases: 'more than' adds to variable (n+5), 'less than' subtracts from variable (n-7), 'times' multiplies (3n coefficient notation), 'of' often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $n$ represent a number. Which expression represents 7 less than $n$ ?
Answer: $n-7$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). Example: "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for "7 less than n" is n - 7. A common error is reversing the order, like choosing 7 - n instead of n - 7, or mistaking it for addition like n + 7 or multiplication like 7n. Writing: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $n$ represent a number. Which expression means 5 more than $n$ ?
Answer: $n+5$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: 'more than/sum' → addition (+), 'less than/difference' → subtraction (-), 'times/product' → multiplication (coefficient: 3x means 3 times x), 'divided by/quotient' → division (x/4). Order matters for non-commutative: '5 less than n' means subtract 5 FROM n (n-5, not 5-n reversed), 'n divided by 4' means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, '7 more than a number' uses variable n for number, 'more than' means add, expression: n+7; 'product of 3 and x' means 3 times x: 3x; 'twice a number plus 5' means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); 'x divided by 8' means x/8. The correct translation for '5 more than n' is n + 5, which matches choice B. A common error is reversing the order, like writing 5 - n instead of n - 5 for subtraction phrases, or confusing 'more than' with multiplication to get 5n. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for 'a number'), (3) determine order (for subtraction/division: 'less than' and 'divided by' require careful order—'5 less than n' is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: '5 more than n' gives 10+5=15✓, makes sense). Key phrases: 'more than' adds to variable (n+5), 'less than' subtracts from variable (n-7), 'times' multiplies (3n coefficient notation), 'of' often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $x$ represent the number of minutes Maya practices piano. Which expression represents 5 more than the number of minutes she practices?
Answer: $x+5$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). Example: "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for "5 more than the number of minutes she practices" where x is the minutes is x + 5. A common error is confusing addition with subtraction, like choosing x - 5 instead of x + 5, or mistaking it for multiplication like 5x. Writing: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $x$ represent a number. Which expression means 7 less than $x$ ?
Answer: $x-7$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: 'more than/sum' → addition (+), 'less than/difference' → subtraction (-), 'times/product' → multiplication (coefficient: $3x$ means 3 times x), 'divided by/quotient' → division ( $x/4$ ). Order matters for non-commutative: '5 less than n' means subtract 5 FROM n ( $n-5$ , not $5-n$ reversed), 'n divided by 4' means $n/4$ (not $4/n$ ). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, '7 more than a number' uses variable n for number, 'more than' means add, expression: $n+7$ ; 'product of 3 and x' means 3 times x: $3x$ ; 'twice a number plus 5' means 2 times n plus 5: $2n+5$ (not $2(n+5)$ which would be twice the sum); 'x divided by 8' means $x/8$ . The correct translation for '7 less than x' is $x - 7$ , which matches choice C. A common error is reversing the order, like writing $7 - x$ instead of $x - 7$ for 'less than' phrases, or mistaking it for addition to get $x + 7$ . To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for 'a number'), (3) determine order (for subtraction/division: 'less than' and 'divided by' require careful order—'5 less than n' is $n-5$ , subtract from first quantity), (4) write expression ( $n+5$ , $3x$ , $x-7$ , $y/4$ ), (5) verify (if n=10: '5 more than n' gives $10+5=15$ ✓, makes sense). Key phrases: 'more than' adds to variable ( $n+5$ ), 'less than' subtracts from variable ( $n-7$ ), 'times' multiplies ( $3n$ coefficient notation), 'of' often means multiply (half of n: $(1/2)n=n/2$ ). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing $3×x$ instead of $3x$ , acceptable but coefficient notation preferred).
Which expression represents the sum of 3 times $x$ and 5?
Answer: $3x+5$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for 'the sum of 3 times x and 5' is 3x + 5, which matches choice B. A common error is misinterpreting as multiplying the sum, like 3(x + 5), or using addition without multiplication to get 3 + x + 5. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
Let $x$ represent the number of stickers Jordan has. Which expression represents 2 more than twice the number of stickers?
Answer: $2x+2$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). Example: "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for "2 more than twice the number of stickers" where x is the number is 2x + 2. A common error is misunderstanding the combined operations, like choosing 2(x + 2) which is twice the sum instead of 2x + 2, or x + 2x which is 3x. Writing: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A movie ticket costs $x$ dollars. You also pay a $3 fee. Which expression represents the total cost?
Answer: $x+3$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for the total cost, which is x dollars plus a $3 fee, is x + 3, which matches choice C. A common error is using subtraction like x - 3, or multiplication like 3x, instead of addition. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A notebook costs $x$ dollars. A student buys 3 notebooks. Which expression represents the total cost in dollars?
Answer: $3x$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: 'more than/sum' → addition (+), 'less than/difference' → subtraction (-), 'times/product' → multiplication (coefficient: 3x means 3 times x), 'divided by/quotient' → division (x/4). Order matters for non-commutative: '5 less than n' means subtract 5 FROM n (n-5, not 5-n reversed), 'n divided by 4' means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, '7 more than a number' uses variable n for number, 'more than' means add, expression: n+7; 'product of 3 and x' means 3 times x: 3x; 'twice a number plus 5' means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); 'x divided by 8' means x/8. The correct translation for the total cost of 3 notebooks at x dollars each is 3x, which matches choice B. A common error is confusing multiplication with addition, like writing x + 3 instead of 3x, or using division like x/3. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for 'a number'), (3) determine order (for subtraction/division: 'less than' and 'divided by' require careful order—'5 less than n' is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: '5 more than n' gives 10+5=15✓, makes sense). Key phrases: 'more than' adds to variable (n+5), 'less than' subtracts from variable (n-7), 'times' multiplies (3n coefficient notation), 'of' often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).
A pack has $y$ stickers. You buy 3 packs. Which expression represents the product of 3 and $y$ ?
Answer: $3y$ This question tests writing algebraic expressions from verbal descriptions using variables (letters representing numbers) and operation symbols (+, -, ×, ÷). Translating words to algebra: "more than/sum" → addition (+), "less than/difference" → subtraction (-), "times/product" → multiplication (coefficient: 3x means 3 times x), "divided by/quotient" → division (x/4). Order matters for non-commutative: "5 less than n" means subtract 5 FROM n (n-5, not 5-n reversed), "n divided by 4" means n/4 (not 4/n). Variable represents number: x, n, y are placeholders (unknown or any number in context). For example, "7 more than a number" uses variable n for number, "more than" means add, expression: n+7; "product of 3 and x" means 3 times x: 3x; "twice a number plus 5" means 2 times n plus 5: 2n+5 (not 2(n+5) which would be twice the sum); "x divided by 8" means x/8. The correct translation for 'product of 3 and y' is 3y, which matches choice C. A common error is confusing product with sum, like writing 3 + y, or using division instead to get y ÷ 3. To write expressions: (1) identify operation words (more→add, times→multiply, less→subtract, divided→divide), (2) choose variable (n, x, y for "a number"), (3) determine order (for subtraction/division: "less than" and "divided by" require careful order—"5 less than n" is n-5, subtract from first quantity), (4) write expression (n+5, 3x, x-7, y/4), (5) verify (if n=10: "5 more than n" gives 10+5=15✓, makes sense). Key phrases: "more than" adds to variable (n+5), "less than" subtracts from variable (n-7), "times" multiplies (3n coefficient notation), "of" often means multiply (half of n: (1/2)n=n/2). Mistakes: reversing order for subtraction/division (most common: less than and divided by backward), wrong operation (confusing sum and product), coefficient unclear (writing 3×x instead of 3x, acceptable but coefficient notation preferred).

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